Cryptic constructions of Hadamard matrices

On August 12, 2026, Levent Alpöge of Anthropic posted on X constructions for Hadamard matrices of orders $n \in \{668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964\}$, resolving all previously unknown orders below 2000 (but leaving the full conjecture open). A matrix $H \in \mathcal{M}^{n \times n}(\left\{ \pm 1\right\})$ is Hadamard of order $n$ if

It is conjectured that such matrices exist for all orders $4n$, where $n$ is any positive integer. This is an open problem that has long intrigued me, and it’s exciting to see progress!

However, the post was, in the unfortunate style du jour, a “vaguepost”: an intentionally cryptic post to bait users into engagement. This is a term that I hadn’t come across before but that I think is apropos (especially in light of the similar Jacobian conjecture resolution).

To me, this was immediately reminiscent of the anonymous maths Stack Exchange user, “Cleo”, who posted closed-form solutions to intricate integral problems without providing any working or reasoning.

There is a sense in which this adds to the allure of mathematics, and perhaps elevates individual endeavours in the public imagination. However, on the whole, I find the attitude tiresome and borderline hostile to the broader community. In particular, there is no context provided on the (likely novel) method of construction.

I have taken the liberty of posting the unobfuscated matrices in plaintext as a gist. To verify it:

$ uvx --with numpy ipython
import numpy as np

for i in (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964):
    H = np.loadtxt(f"matrices/hadamard_{i}.txt")
    n = H.shape[0]
    assert np.array_equal(H @ H.T, n * np.eye(n))